Normal Space - Definitions

Definitions

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.

A T₄ space is a T₁ space X that is normal; this is equivalent to X being Hausdorff and normal.

A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.

A completely T₄ space, or T₅ space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T₄ space.

A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval .)

It turns out that X is perfectly normal if and only if X is normal and every closed set is a G_δ set. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.

A Hausdorff perfectly normal space X is a T₆ space, or perfectly T₄ space.

Note that the terms "normal space" and "T₄" and derived concepts occasionally have a different meaning. (Nonetheless, "T₅" always means the same as "completely T₄", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.

Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature – they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T₄ space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T₄", or "completely normal Hausdorff" instead of "T₅".

Fully normal spaces and fully T₄ spaces are discussed elsewhere; they are related to paracompactness.

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.